Hi Tom,
> For a simple (one-page) proof of the result sought below in ZF without
> Choice see p. 550 of the article in The American Mathematical Monthly
> (MAA),vol 109, No. 6, (June-July, 2002) by Leonard Gillman (pp544-553)
>> "Two Classical Surprises Concerning the Axiom
> of Choice and the Continuum Hypothesis"
>> best regards,
> Tom Dunion
The question (whether aleph(X) injects into P(P(X))) is somewhat more
subtle than this. In Gillman's paper two results that seem related to my
question are shown. The first is theorem 1, that aleph(X) injects into
P(P(P(X))). The second is lemma 3, which gives that if aleph(X) injects
into P(P(P(X))), then it also injects into P(P(X)). Unfortunately, lemma 3
also implies that X is well-orderable, and hence it is not a theorem of
ZF, which makes sense, since it is easy to give counterexamples to lemma 3
in ZF, and Gillman's proof uses GCH.
This is what I know at the moment about the problem (assume X is
infinite): Let aleph_*(X) be the supremum of the ordinals onto which X can
be mapped, so aleph(X)\le aleph_*(X) and strict inequality is possible in
the absence of choice.
* P(aleph(X)) injects P(P(X^2)).
* In particular, P(alpha^+) injects into P^2(alpha) for alpha an infinite
ordinal.
* Also, since aleph(Y)=aleph(Y^2) for any infinite Y, if X is a square
(i.e., equipotent to Y^2 for some Y), then P(aleph(X)) injects into
P^2(X).
* If alpha is below aleph_*(X), then P^2(alpha) injects into P^2(X).
* In particular, P(aleph(X)) injects into P^2(X) if aleph(X) is a
successor cardinal or aleph(X)<aleph_*(X).
* If aleph(X)=kappa and kappa is the kappa-th initial ordinal, then
P(aleph(X)) injects into P^2(X).
* In particular, P(aleph(X)) injects into P^2(X) if X is Dedekind-finite.
Hence, if there is an X with aleph(X)=aleph(P^2(X)), then necessarily
(X is not a square and) aleph(X)=aleph_*(X)=aleph_lambda for some infinite
limit ordinal lambda < aleph_lambda.
All these are easy observations. At this point, I think the question is
open, but I would be grateful for any suggestions or references.
Best,
Andres